Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 7, Issue 4, Article 126, 2006
ON AN UPPER BOUND FOR THE DEVIATIONS FROM THE MEAN VALUE
AURELIA CIPU
R AILWAY H IGHSCHOOL B UCHAREST
S TR . B UTUCENI , NR .10
B UCHAREST, ROMANIA
aureliacipu@netscape.net
Received 12 October, 2005; accepted 20 February, 2006
Communicated by A. Sofo
A BSTRACT. A completely elementary proof of a known upper bound for the deviations from the
mean value is given. Related inequalities are also discussed. Applications to triangle inequalities
provide characterizations of isosceles triangles.
Key words and phrases: Arithmetic mean, Square mean, Cauchy-Schwarz-Buniakovski inequality, Triangle inequality.
2000 Mathematics Subject Classification. 26D15, 51M16.
1. I NTRODUCTION
For positive real numbers x1 , x2 , . . . , xn , n ≥ 2, one denotes by a their arithmetic mean and
by b the arithmetic mean of their squares (also known as the square mean of the given numbers).
In [1], V. Nicula proves the inequalities
p
(1.1)
|xk − a| ≤ (n − 1)(b − a2 ) ,
k = 1, 2, . . . , n ,
which are equivalent to
(1.2)
max{ |xk − a| : 1 ≤ k ≤ n } ≤
p
(n − 1)(b − a2 ) .
The proof uses calculus and the author asks for an elementary proof. The aim of this paper is to provide such an approach. Our proof uses nothing more than the Cauchy-SchwarzBuniakovski (CSB for short) inequality and is therefore accessible to pupils acquainted with
polynomials of degree two. This approach has an additional advantage—it makes it very easy
to determine the necessary and sufficient conditions under which equality holds in (1.2).
In the next section we shall prove the result below.
Theorem 1.1. Let n > 1 be an integer and x1 , x2 , . . . , xn be positive real numbers. Denote
a = (x1 + x2 + · · · + xn )/n and b = (x21 + x22 + · · · + x2n )/n. Then
p
max{ |xk − a| : 1 ≤ k ≤ n } ≤ (n − 1)(b − a2 ) .
ISSN (electronic): 1443-5756
c 2006 Victoria University. All rights reserved.
308-05
2
AURELIA C IPU
The equality holds in this relation if and only if either n = 2 or n ≥ 3 and n − 1 of the given
numbers are equal.
Section 3 contains several consequences of this result or of its proof. In the final section
we give applications to triangle inequalities. In addition to being new, these results provide
characterizations for isosceles triangles.
2. A S IMPLE P ROOF
Let us denote by yk := xk − a, k = 1, 2, . . . , n, the deviations from the mean value. Then
we have
n
X
(2.1)
yk =
n
X
k=1
(2.2)
n
X
k=1
yk2
=
n
X
xk − na = 0,
k=1
x2k
− 2a
k=1
n
X
xk + na2 = n(b − a2 ).
k=1
From equation (2.2) it follows that b ≥ a2 , so the square root in the statement of Theorem 1.1
is real.
We have
!2
n−1
X
by (2.1)
yn2 = −
yk
k=1
n−1
X
≤ (n − 1)
yk2
by CSB
k=1
= (n − 1)
n
X
!
yk2
−
yn2
k=1
= n(n − 1)(b − a2 ) − (n − 1)yn2
by (2.2).
Hence,
yn2 ≤ (n − 1)(b − a2 ).
(2.3)
Taking the square root results in relation (1.1) written for k = n. A similar reasoning yields the
inequalities for k = 1, 2, . . . , n − 1.
Let us determine when equality holds in relation (1.2). It is easily seen that for n = 2 we
have
√
1
| x1 − a |=| x2 − a |= | x1 − x2 |= b − a2 .
2
For n ≥ 3, the equality holds in relation (2.3) if and only if the CSB inequality for y1 , y2 , . . . ,
yn−1 turns into an equality. It is well-known that this is the case if and only if all the involved
numbers coincide. In terms of the given numbers, the necessary and sufficient condition that
xn − a = (n − 1)(b − a2 )
is x1 = x2 = · · · = xn−1 .
Theorem 1.1 is proved.
J. Inequal. Pure and Appl. Math., 7(4) Art. 126, 2006
http://jipam.vu.edu.au/
O N AN U PPER B OUND FOR THE D EVIATIONS FROM THE M EAN VALUE
3
3. C ONSEQUENCES
The reasoning used to characterize the equality in relation (1.2) immediately yields:
Corollary 3.1. If equality holds in relation (1.1) for two values of the index k, then all numbers
are equal.
Inequality (1.2) has a companion inequality, in which the maximal deviation from the mean
value is bounded from below in terms of a and b.
Corollary 3.2. Using the same hypothesis and notation we have
√
max{ |xk − a| : 1 ≤ k ≤ n } ≥ b − a2 .
The equality holds in this relation if and only if b = a2 , that is, when all xk coincide.
A natural question is whether there are similar results for the smallest deviation from the
mean value. From relation (2.2) one easily gets an upper bound.
√
Corollary 3.3. min{ |xk − a| : 1 ≤ k ≤ n } ≤ b − a2 .
However, in general there are no lower bounds for the smallest deviation from the mean value
better than the trivial one (being fulfilled by the absolute value of any expression)
min{ |xk − a| : 1 ≤ k ≤ n } ≥ 0.
Indeed, the claim is clear if one considers n numbers, one of which is equal to the arithmetic
mean of the others.
A final remark compares inequality (1.2) to inequalities obtained by other methods. When
xk = k, k = 1, 2, . . . , n, we have a = (n + 1)/2 and b = (n + 1)(2n + 1)/6, and Theorem 1.1
yields
r
n
+
1
n
−
1
n+1
: 1≤k≤n ≤
(3.1)
max k −
.
2
2
3
On the other hand, since xk are contained in an interval of length n − 1, from the geometric
interpretation of the modulus one obtains the stronger inequality
n
+
1
: 1 ≤ k ≤ n ≤ n − 1.
max k −
2 2
4. A PPLICATIONS
The previous results allow us to characterize isosceles triangles. The idea is to conveniently
specialize the xk in Theorem 1.1. The computations needed in each case are simple and therefore omitted. Applying Theorem 1.1 successively for n = 3 and xk the side lengths, altitudes,
and radii of excircles, one obtains
Proposition 4.1. In any triangle with sides of length a, b, c, one denotes by s = (a + b + c)/2 its
semiperimeter, by F its area, and by r and R the radius of the in- and circumcircle, respectively.
Then
√
max |b + c − 2a| ≤ 2 a2 + b2 + c2 − ab − bc − ca ,
√
max |ab + bc − ca| ≤ 2 a2 b2 + b2 c2 + c2 a2 − 4sabc ,
p
3F max r + 4R −
≤ 2 (r + 4R)2 − s2 .
s−a
In each relation the equality holds if and only if the triangle is isosceles.
J. Inequal. Pure and Appl. Math., 7(4) Art. 126, 2006
http://jipam.vu.edu.au/
4
AURELIA C IPU
These results have an additional interest due to the fact that there are comparatively few
symmetrical inequalities in which equality holds not only when all variables are equal.
R EFERENCES
[1] V. NICULA, On a classical extremum problem (Romanian), Gaz. Matem. (Bucharest), 100 (1995),
498–501.
J. Inequal. Pure and Appl. Math., 7(4) Art. 126, 2006
http://jipam.vu.edu.au/